OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
171 
must determine whether the translations of v from A to A', from A' to A", from A" to 
A"', &c. are equivalent to each other in effect; for, if they are, it makes no difference 
in effect whether we repeat the translation of v from A to A' five times, or simply 
translate v from A to A', from A' to A", from A" to A'", and so on to B. The question 
then comes to this — Are we to regard translations as equivalent to each other, in 
effect, when the magnitudes translated and the lines along which they are translated 
are respectively equivalent to each other, whether the translations take place m the 
same part of space or not ? 
(29.) Fundamental Assumption. — I am thus led to make the following Assumption 
the basis of my proposed method; viz. — That parallel and equal 
translations of parallel and equal magnitudes are equivalent to each 
other both as regards the lateral and longitudinal effects ; or, sym- 
bolically, if u and y' be respectively parallel and equal to u and v, ^ 
then 
Fig. 17. 
'll' 
vl .v—u.v, and u' Xv'=uXv. 
This assumption holds true, manifestly, in each of the three suggesting cases from 
which I have taken my start (see art. 2), and I am therefore justified in adopting it, 
with the understanding, of course, that it be shown to hold true, or tacitly admitted, 
in all cases to which the notation may be applied ; or else, should the occasion 
require it, be abandoned, and, with it, the property expressed by the equation {mu).v 
=m{u.v). 
(30.) Returning to fig. 16, we have, by the Assumption just made, 
AA' .v= A' A” .v= A” A'" .v= &c. &c. ; 
.'. m{u.v) = m{AA' .v)=AA' .v-\-A'A" . v-\-A"A''' . v-{- &c. 
= ( A A' + A' A" + A" A'" + &c.) . y 
= {mu) .V. 
And generally, by what has been proved, we have 
{mu) .{nv) = n{{mu) .v] =:mn{u .v ) ; 
and, similarly, {mu) X {nv) =mn{u X v). 
It appears thus that numerical coeff dents, occurring in the symbolic forms u.v and 
uXv, may always be brought out and incorporated by actual multiplication*. 
(31.) May the order of the factors u and v in the symbolic forms u.v and uXv 
changed, or not) — First, as regards u.v, may we put u.v=v.u} Here I may repeat 
that the second factor always denotes the translated magnitude, or rather, the repre- 
sentative arrow. Thus the question is — as regards lateral effect, is the translation 
of the magnitude represented by the arrow v along the line u equivalent to the trans- 
lation of that represented by the arrow u along the line v. This is easily decided as 
follows. 
(32.) It is clear that the lateral effect of the translation of a magnitude in its own 
* It may be shown, in the usual way, that this is true also when m and n axe fractional or negative numbers. 
z 2 
